Chapter 1.5, Problem 33E

Rewrite each of these statements so that negations ap

   $\neg$ pear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

a) $\neg \forall x\forall yP\left(x,y\right)$

b) $\neg \forall y\exists xP\left(x,y\right)$

c) $\neg \forall y\forall x\left(P\left(x,y\right)\vee Q\left(x,y\right)\right)$

d) $\neg \left(\exists x\exists y\neg P\left(x,y\right)\wedge \forall x\forall yQ\left(x,y\right)\right)$

e) $\neg \forall x\left(\exists y\forall zP\left(x,y,z\right)\wedge \exists z\forall yP\left(x,y,z\right)\right)$